Mathematical Sciences: p-adic Automorphic Forms

这项研究涉及 p 进对称空间、模形式的算术以及数论和代数几何中的一些计算和算法问题的研究。在p-adic自同构形式的一般领域中，关于解析函数和p-adic上半平面上的测度及其边界上的测度之间关系的已知结果被推广到Drinfeld的高维p-adic上半空间。这些结果对由 Drinfeld 空间统一的代数簇的几何具有影响，包括某些 Shimura 簇和 Drinfeld 模簇。它们对于一般线性群的 p 进表示理论也有重要的影响。在模形式领域，主要研究者正在继续研究与“异常零猜想”相关的问题。最后，对计算 Picard-Fuchs 微分方程和高斯-马宁连接的算法的早期工作进行了扩展。 该项目的研究涉及算术几何和自守形式的一般领域。代数几何是现代数学最古老的部分之一。在过去的十年里，它已经蓬勃发展，解决了几个世纪以来的问题。最初，它处理由最简单的方程（即多项式）定义的平面中的图形。如今，该领域不仅使用代数方法，还使用分析和拓扑方法。相反，它广泛应用于这些领域。此外，它已被证明在物理学、理论计算机科学、密码学、编码理论和机器人技术等各个领域都很有用。自守形式起源于十九世纪中叶的非欧几里得几何。数学家和物理学家很早就认识到，许多具有根本重要性的物体的基本性质都是非欧几里得的。该领域主要关注有关整数的问题，但在几何和分析的使用中，它保留了与其历史根源的联系，从而与理论物理学中的规范理论和信息论中的编码理论等不同领域的问题保持联系。 这项研究涉及 p 进对称空间、模形式的算术以及数论和代数几何中的一些计算和算法问题的研究。在p-adic自同构形式的一般领域中，关于解析函数和p-adic上半平面上的测度及其边界上的测度之间关系的已知结果被推广到Drinfeld的高维p-adic上半空间。这样的结果讨厌对由德林菲尔德空间统一的代数簇的几何意义，包括某些志村簇和德林菲尔德模簇。它们对于一般线性群的 p 进表示理论也有重要的影响。在模形式领域，主要研究者正在继续研究与“异常零猜想”相关的问题。最后，对计算 Picard-Fuchs 微分方程和高斯-马宁连接的算法的早期工作进行了扩展。 该项目的研究涉及算术几何和自守形式的一般领域。代数几何是现代数学最古老的部分之一。在过去的十年里，它已经蓬勃发展，解决了几个世纪以来的问题。最初，它处理由最简单的方程（即多项式）定义的平面中的图形。如今，该领域不仅使用代数方法，还使用分析和拓扑方法。相反，它广泛应用于这些领域。此外，它已被证明在物理学、理论计算机科学、密码学、编码理论和机器人技术等各个领域都很有用。自守形式起源于十九世纪中叶的非欧几里得几何。数学家和物理学家很早就认识到，许多具有根本重要性的物体的基本性质都是非欧几里得的。该领域主要关注有关整数的问题，但在几何和分析的使用中，它保留了与其历史根源的联系，从而与理论物理学中的规范理论和信息论中的编码理论等不同领域的问题保持联系。

This research involves the study of p-adic symmetric spaces, the arithmetic of modular forms, and a number of computational and algorithmic questions in number theory and algebraic geometry. In the general area of p-adic automorphic forms, known results about the relationship between analytic functions and measures on the p-adic upper half plane and measures on its boundary are generalized to Drinfeld's higher dimensional p-adic upper half spaces. Such results have implications for the geometry of algebraic varieties which are uniformized by the Drinfeld spaces, including certain Shimura varieties and Drinfeld modular varieties. They also have important consequences for the p-adic representation theory of the general linear group. In the area of modular forms, the principal investigator is continuing to study questions relating to the "exceptional zero conjecture". Finally, earlier work on algorithms for computing Picard-Fuchs differential equations and the Gauss-Manin connection are extended. The research in this project lies in the general areas of arithmetic geometry and automorphic forms. Algebraic geometry is one of the oldest parts of modern mathematics. In the past ten years, it has blossomed to the point where it has solved problems that have stood for centuries. Originally, it treated figures in the plane defined by the simplest of equations, namely polynomials. Today, the field utilizes methods not only from algebra, but also from analysis and topology; conversely, it is extensively used in those fields. Moreover, it has proved itself useful in fields as diverse as physics, theoretical computer science, cryptography, coding theory and robotics. Automorphic forms arose out of non-Euclidean geometry in the middle of the nineteenth century. Both mathematicians and physicists have thus long realized that many objects of fundamental importance are non-Euclidean in their basic nature. This field is principally concerned with questions about the whole numbe rs, but in its use of geometry and analysis, it retains connection to its historical roots and thus to problems in areas as diverse as gauge theory in theoretical physics and coding theory in information theory. This research involves the study of p-adic symmetric spaces, the arithmetic of modular forms, and a number of computational and algorithmic questions in number theory and algebraic geometry. In the general area of p-adic automorphic forms, known results about the relationship between analytic functions and measures on the p-adic upper half plane and measures on its boundary are generalized to Drinfeld's higher dimensional p-adic upper half spaces. Such results hate implications for the geometry of algebraic varieties which are uniformized by the Drinfeld spaces, including certain Shimura varieties and Drinfeld modular varieties. They also have important consequences for the p-adic representation theory of the general linear group. In the area of modular forms, the principal investigator is continuing to study questions relating to the "exceptional zero conjecture". Finally, earlier work on algorithms for computing Picard-Fuchs differential equations and the Gauss-Manin connection are extended. The research in this project lies in the general areas of arithmetic geometry and automorphic forms. Algebraic geometry is one of the oldest parts of modern mathematics. In the past ten years, it has blossomed to the point where it has solved problems that have stood for centuries. Originally, it treated figures in the plane defined by the simplest of equations, namely polynomials. Today, the field utilizes methods not only from algebra, but also from analysis and topology; conversely, it is extensively used in those fields. Moreover, it has proved itself useful in fields as diverse as physics, theoretical computer science, cryptography, coding theory and robotics. Automorphic forms arose out of non-Euclidean geometry in the middle of the nineteenth century. B oth mathematicians and physicists have thus long realized that many objects of fundamental importance are non-Euclidean in their basic nature. This field is principally concerned with questions about the whole numbers, but in its use of geometry and analysis, it retains connection to its historical roots and thus to problems in areas as diverse as gauge theory in theoretical physics and coding theory in information theory.